Residual analysis of 100% fit model using system identification toolbox

I have used the system identification toolbox to estimate a simple transfer function representing a synchronous machine swing equation feedback system that is input rotational speed/frequency and outputs a change in power. I know the exact transfer function and that it has two poles and two zeros. The purpose of the study is to trial the toolbox on a known transfer function before using it for unknown systems.

I am interested in how the system responds to different changes in frequency, so have simulated the output response to several frequency scenarios (in simulink) and have tried to estimate models from each of the scenarios. The estimation tool achieves a 100% fit to measured outputs when trained on each frequency scenario input and validated against the same and different frequency scenarios.

However, the residual analysis of the models is not satisfying the criteria used to determine the good quality of the models. The whiteness criteria requires the residuals to be uncorrelated with themselves, but the residuals shown have high autocorrelation. Included is the residual analysis for two models for a given frequency input : the estimation model which was trained on the same frequency input scenario and the validation model which was trained on a different frequency input.

Residuals represent the difference between the modelled and measured outputs. So I understand the residuals of a model that exactly fits the measurements would be zero. Then the autocorrelation should be 1 for a model which has the same residual at all times. Is this an appropriate reason to ignore the normal threshold limits that qualify a good model? The estimation model has a residual autocorrelation almost equal to 1 (within 1e-2) for all lags - can i assume the proximity to 1 results from the 100% fit?

What about the validation model, which is trained on a different frequency input dataset but also has 100% fit for the output to this input? Why does it not have a constant residual autocorrelation of 1?

Finally, the independence test uses the cross correlation between input and residuals to ensure that the residuals are not related to the previous inputs. The validation model residual has high correlation with negative (future) inputs due to the feedback nature of the system, which is fine, but does the relation to past inputs (positive lag) matter considering the 100% fit?

Thanks

Answer 1

[Rajiv Singh]

With simulated data with no noise, it is difficult to read the residual results since there is no baseline noise floor. That is, the residual are so small enough in absolute terms that there relative (correlation) analysis is not meaningful. For more realistic residual analysis, add some noise to the simulated output.

Perhaps you can post an example? The example can include just with the estimation and validation datasets and information on the expected order of the model.

[Sam Harrison]

I have attached a figure with subplots of the frequency change input to the synchronous machine and the power change output I want to replicate with the estimated model.

I have also attached a figure of the output of the 100% fit tf model on top of the measured power change output and a figure of the residual analysis for this model.

The transfer function takes the form: where is the damping factor, is the synchronising torque, is the synchronous frequency and H is the inertial time constant. From this I set the estimated model to have 2 poles and 2 zeros.

Let me know if you need any more information.

Thanks